I do the research on a difference of players' behaviour on different levels. There is no problem to find games with players rated over 2000 ELO but I lack those below 2000. Does anybody know a source with such games?
You can find a large collection of games played on FICS archived from the superb site: ficsgames.org Here you can find downloads of games based on rating category and type of games.
Edit: Now we also have LiChess giving out all games! Check out database.lichess.org. Even though it is not already categorized into Elo, you can easily do it via a pgn parser program.
The reason nobody has gone to the trouble of collecting games of sub 2000 players is because the quality is not expected to be high and not because they are not available.
Of course the FIDE rating rules also mean that fewer of these games are saved in the first place. For all events where norms are available the arbiter must send in to FIDE a pgn file with all the games. This is for the obvious reason of preventing fraud with title norms.
For FIDE rated tournaments where norms are not available (in other words the ones with few higher rated players and lots of lower rated ones) sending in this file to FIDE is optional. Still some arbiters do it although many, maybe most, do not because it is a lot of (unpaid) work.
So, these games are out there. You will find them for individual tournaments on the FIDE rating site and sites like chess-results.com (because a large number of arbiters use the Swiss Manager pairing program which works with chess-results.com). I believe sites like chess-db.com periodically look at these sites and collect games. You can find games for named individuals by searching the chess-db.com site.
In short nobody has done the hard work for you of collecting all these games into a sub-2000 database but games are out there if you want to do the work yourself.